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The sigmoid function is strictly increasing. Therefore the risky region is $\hat{\eta }(x)\ge \log \{p/(1-p)\}$. Its decision boundary is the hyperplanePoints on or above this score threshold are assigned to the risky class. For $p=1/2$, the right side is zero.

The decision boundary is the hyperplaneAt threshold $q$ the classifier predicts spam whenso $q=1/2$ gives the original classifier. Varying $q$ translates the boundary parallel to itself: raising $q$ shrinks the region classified as spam and generally trades fewer false positives for more false negatives.

Let $N_k(x)$ index the $k$ closest training covariates to $x$. The K-nearest neighbors algorithm estimatesand predicts one when this average is at least $1/2$.

Small $k$ gives low smoothing bias but high sampling variance and a jagged decision boundary. Larger $k$ averages more labels, reducing variance and producing a smoother boundary, but it mixes increasingly distant covariates and raises bias. The optimal balance depends on sample size, dimension, and smoothness of ${\eta }_1$, and is commonly selected by cross-validation.

An unpruned maximal CART classifier repeatedly cuts with vertical or horizontal lines until every terminal rectangle is pure or contains observations that cannot be separated by an axis-aligned split. Here the distinct grid points can be isolated into pure rectangles, producing a step-shaped, axis-aligned decision boundary and zero training error.